Required Libraries¶
In [ ]:
import yfinance as yf
import pandas as pd
from datetime import datetime
from arch import arch_model
import matplotlib.pyplot as plt
import numpy as np
import warnings
warnings.filterwarnings("ignore")
Question 1¶
In [ ]:
start_date = "2022-08-30"
end_date = "2024-09-30"
In [ ]:
exchange_tickers = ["ZAR=X", "AUDUSD=X", "GBPUSD=X"]
exchange_data = {ticker: yf.download(ticker, start=start_date, end=end_date) for ticker in exchange_tickers}
[*********************100%%**********************] 1 of 1 completed [*********************100%%**********************] 1 of 1 completed [*********************100%%**********************] 1 of 1 completed
In [ ]:
for ticker, data in exchange_data.items():
print(f"Data for {ticker}")
print(data.head())
Data for ZAR=X
Open High Low Close Adj Close Volume
Date
2022-08-30 16.809641 16.972799 16.745100 16.809641 16.809641 0
2022-08-31 16.957800 17.087179 16.858629 16.957800 16.957800 0
2022-09-01 17.105209 17.304930 17.088301 17.105209 17.105209 0
2022-09-02 17.251900 17.345699 17.172701 17.251900 17.251900 0
2022-09-05 17.300900 17.418039 17.146500 17.300900 17.300900 0
Data for AUDUSD=X
Open High Low Close Adj Close Volume
Date
2022-08-30 0.690751 0.695797 0.685990 0.690751 0.690751 0
2022-08-31 0.685740 0.690422 0.684340 0.685740 0.685740 0
2022-09-01 0.682850 0.684360 0.677172 0.682850 0.682850 0
2022-09-02 0.679530 0.685380 0.678109 0.679530 0.679530 0
2022-09-05 0.679030 0.680411 0.677360 0.679030 0.679030 0
Data for GBPUSD=X
Open High Low Close Adj Close Volume
Date
2022-08-30 1.172127 1.175807 1.162304 1.172251 1.172251 0
2022-08-31 1.166086 1.169454 1.160025 1.165977 1.165977 0
2022-09-01 1.159689 1.161710 1.150324 1.159851 1.159851 0
2022-09-02 1.154734 1.158856 1.153136 1.154894 1.154894 0
2022-09-05 1.147855 1.152074 1.144518 1.147460 1.147460 0
In [ ]:
def calculate_returns(df):
df['Returns'] = df['Adj Close'].pct_change().dropna() * 100
return df['Returns'].dropna()
In [ ]:
def fit_garch_model(returns, model_type='GARCH'):
if model_type == 'ARCH':
model = arch_model(returns, vol='Arch', p=1)
elif model_type == 'GARCH':
model = arch_model(returns, vol='Garch', p=1, q=1)
elif model_type == 'EGARCH':
model = arch_model(returns, vol='EGarch', p=1, q=1)
elif model_type == 'TGARCH':
model = arch_model(returns, vol='Garch', p=1, o=1, q=1)
elif model_type == 'PGARCH':
model = arch_model(returns, p=1, q=1, o=1, power=1.0)
else:
raise ValueError(f"Unsupport model type: {model_type}")
model_fit = model.fit(disp='off')
return model_fit
In [ ]:
def visualize_results(returns, model_results, title):
# plot returns
plt.figure(figsize=(20, 10))
plt.subplot(2, 1, 1)
plt.plot(returns, label='Returns', color='blue')
plt.title(f"Daily Returns for {title}", fontsize=20, fontweight='bold')
plt.xlabel('Date')
plt.ylabel('Returns')
plt.legend()
# plot volatility
plt.subplot(2, 1, 2)
plt.plot(model_results.conditional_volatility, label='Conditional Volatility', color='red')
plt.title(f"Conditional Volatility for {title}", fontsize=20, fontweight='bold')
plt.xlabel('Date')
plt.ylabel('Volatility')
plt.legend()
plt.tight_layout()
plt.show()
In [ ]:
def compare_volatility_forecasting(returns, models):
realized_volatility = returns**2
forecast_errors = {}
for model_name, model_results in models.items():
forecasted_volatility = model_results.conditional_volatility
mse = np.mean((realized_volatility - forecasted_volatility**2)**2)
forecast_errors[model_name] = mse
print(f"{model_name} - Volatility Forecasting MSE: {mse}")
best_model = min(forecast_errors, key=forecast_errors.get)
return best_model
In [ ]:
def evaluate_and_visualize_models(returns, key):
print(f"\n**********************************************************Processing {key}*****************************************************************")
models = {}
# ARCH
print("Fitting ARCH(1) model...")
arch_res = fit_garch_model(returns, model_type='ARCH')
models['ARCH'] = arch_res
print(arch_res.summary())
visualize_results(returns, arch_res, f"{key} ARCH(1)")
# GARCH
print("Fitting GARCH(1,1) model...")
garch_res = fit_garch_model(returns, model_type='GARCH')
models['GARCH'] = garch_res
print(garch_res.summary())
visualize_results(returns, garch_res, f"{key} GARCH(1,1)")
# EGARCH
print("Fitting EGARCH(1,1) model...")
egarch_res = fit_garch_model(returns, model_type='EGARCH')
models['EGARCH'] = egarch_res
print(egarch_res.summary())
visualize_results(returns, egarch_res, f"{key} EGARCH(1,1)")
# TGARCH
print("Fitting TGARCH(1,1) model...")
tgarch_res = fit_garch_model(returns, model_type='TGARCH')
models['TGARCH'] = tgarch_res
print(tgarch_res.summary())
visualize_results(returns, tgarch_res, f"{key} TGARCH(1,1)")
# PGARCH
print("Fitting PGARCH(1,1) model...")
pgarch_res = fit_garch_model(returns, model_type='PGARCH')
models['PGARCH'] = pgarch_res
print(pgarch_res.summary())
visualize_results(returns, pgarch_res, f"{key} PGARCH(1,1)")
# Compare AIC and BIC
print("\nModel Comparisons:")
print(f"ARCH AIC: {arch_res.aic}, BIC: {arch_res.bic}")
print(f"GARCH AIC: {garch_res.aic}, BIC: {garch_res.bic}")
print(f"EGARCH AIC: {egarch_res.aic}, BIC: {egarch_res.bic}")
print(f"TGARCH AIC: {tgarch_res.aic}, BIC: {tgarch_res.bic}")
print(f"PGARCH AIC: {pgarch_res.aic}, BIC: {pgarch_res.bic}")
best_bic = min(arch_res.bic, garch_res.bic, egarch_res.bic, tgarch_res.bic, pgarch_res.bic)
if best_bic == arch_res.bic:
print("\nBest model based on BIC: ARCH")
elif best_bic == garch_res.bic:
print("\nBest model based on BIC: GARCH")
elif best_bic == egarch_res.bic:
print("\nBest model based on BIC: EGARCH")
elif best_bic == pgarch_res.bic:
print("\nBest model based on BIC: PGARCH")
else:
print("\nBest model based on BIC: TGARCH")
# Compare models based on volatility forecasting
print("\nComparing models based on volatility forecasting...")
best_forecast_model = compare_volatility_forecasting(returns, models)
print(f"\nBest model based on volatility forecasting (lowest mse): {best_forecast_model}")
In [ ]:
for key, df in exchange_data.items():
returns = calculate_returns(df)
evaluate_and_visualize_models(returns, key)
**********************************************************Processing ZAR=X*****************************************************************
Fitting ARCH(1) model...
Constant Mean - ARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: ARCH Log-Likelihood: -679.737
Distribution: Normal AIC: 1365.47
Method: Maximum Likelihood BIC: 1378.28
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:39 Df Model: 1
Mean Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
mu 0.0134 3.825e-02 0.351 0.726 [-6.157e-02,8.839e-02]
Volatility Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
omega 0.7374 6.009e-02 12.273 1.270e-34 [ 0.620, 0.855]
alpha[1] 0.0421 4.723e-02 0.892 0.373 [-5.046e-02, 0.135]
===========================================================================
Covariance estimator: robust
Fitting GARCH(1,1) model...
Constant Mean - GARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GARCH Log-Likelihood: -676.275
Distribution: Normal AIC: 1360.55
Method: Maximum Likelihood BIC: 1377.63
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:39 Df Model: 1
Mean Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
mu 0.0111 3.791e-02 0.292 0.770 [-6.322e-02,8.539e-02]
Volatility Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
omega 2.3413e-07 2.277e-03 1.028e-04 1.000 [-4.462e-03,4.462e-03]
alpha[1] 8.0898e-07 3.189e-03 2.537e-04 1.000 [-6.249e-03,6.251e-03]
beta[1] 0.9993 1.678e-03 595.342 0.000 [ 0.996, 1.003]
=============================================================================
Covariance estimator: robust
Fitting EGARCH(1,1) model...
Constant Mean - EGARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: EGARCH Log-Likelihood: -670.801
Distribution: Normal AIC: 1349.60
Method: Maximum Likelihood BIC: 1366.68
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:40 Df Model: 1
Mean Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
mu 0.0149 1.210e-08 1.231e+06 0.000 [1.489e-02,1.489e-02]
Volatility Model
==============================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------------
omega 6.6501e-04 3.632e-12 1.831e+08 0.000 [6.650e-04,6.650e-04]
alpha[1] -0.0466 9.617e-09 -4.849e+06 0.000 [-4.664e-02,-4.664e-02]
beta[1] 0.9979 5.895e-10 1.693e+09 0.000 [ 0.998, 0.998]
==============================================================================
Covariance estimator: robust
WARNING: The optimizer did not indicate successful convergence. The message was Iteration limit reached.
See convergence_flag.
c:\Users\Alli Ajagbe\AppData\Local\Programs\Python\Python311\Lib\site-packages\arch\univariate\base.py:766: ConvergenceWarning: The optimizer returned code 9. The message is: Iteration limit reached See scipy.optimize.fmin_slsqp for code meaning. warnings.warn(
Fitting TGARCH(1,1) model...
Constant Mean - GJR-GARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GJR-GARCH Log-Likelihood: -676.178
Distribution: Normal AIC: 1362.36
Method: Maximum Likelihood BIC: 1383.70
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:40 Df Model: 1
Mean Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
mu 6.0180e-03 4.211e-02 0.143 0.886 [-7.652e-02,8.856e-02]
Volatility Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
omega 7.4023e-07 2.767e-03 2.675e-04 1.000 [-5.423e-03,5.424e-03]
alpha[1] 2.9164e-07 1.359e-02 2.147e-05 1.000 [-2.663e-02,2.663e-02]
gamma[1] 6.3935e-03 2.559e-02 0.250 0.803 [-4.376e-02,5.655e-02]
beta[1] 0.9960 4.780e-03 208.361 0.000 [ 0.987, 1.005]
=============================================================================
Covariance estimator: robust
Fitting PGARCH(1,1) model...
Constant Mean - TARCH/ZARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: TARCH/ZARCH Log-Likelihood: -678.950
Distribution: Normal AIC: 1367.90
Method: Maximum Likelihood BIC: 1389.25
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:41 Df Model: 1
Mean Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
mu 0.0138 3.877e-02 0.357 0.721 [-6.216e-02,8.983e-02]
Volatility Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
omega 0.4107 0.169 2.426 1.528e-02 [7.883e-02, 0.742]
alpha[1] 0.0789 4.821e-02 1.638 0.101 [-1.553e-02, 0.173]
gamma[1] -0.0350 5.830e-02 -0.600 0.549 [ -0.149,7.931e-02]
beta[1] 0.4822 0.193 2.494 1.264e-02 [ 0.103, 0.861]
===========================================================================
Covariance estimator: robust
Model Comparisons:
ARCH AIC: 1365.4739414081632, BIC: 1378.281230259282
GARCH AIC: 1360.5508589318451, BIC: 1377.6272440666703
EGARCH AIC: 1349.6025407396355, BIC: 1366.6789258744607
TGARCH AIC: 1362.3556485727302, BIC: 1383.7011299912615
PGARCH AIC: 1367.9000905762286, BIC: 1389.24557199476
Best model based on BIC: EGARCH
Comparing models based on volatility forecasting...
ARCH - Volatility Forecasting MSE: 1.1501724653014256
GARCH - Volatility Forecasting MSE: 1.1363881133247706
EGARCH - Volatility Forecasting MSE: 1.1138408425359607
TGARCH - Volatility Forecasting MSE: 1.135994053682568
PGARCH - Volatility Forecasting MSE: 1.1465230503358819
Best model based on volatility forecasting (lowest mse): EGARCH
**********************************************************Processing AUDUSD=X*****************************************************************
Fitting ARCH(1) model...
Constant Mean - ARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: ARCH Log-Likelihood: -542.644
Distribution: Normal AIC: 1091.29
Method: Maximum Likelihood BIC: 1104.10
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:41 Df Model: 1
Mean Model
==============================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------------
mu -3.5260e-03 2.996e-02 -0.118 0.906 [-6.224e-02,5.519e-02]
Volatility Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
omega 0.4363 3.882e-02 11.237 2.673e-29 [ 0.360, 0.512]
alpha[1] 0.0490 6.578e-02 0.744 0.457 [-7.996e-02, 0.178]
===========================================================================
Covariance estimator: robust
Fitting GARCH(1,1) model...
Constant Mean - GARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GARCH Log-Likelihood: -518.640
Distribution: Normal AIC: 1045.28
Method: Maximum Likelihood BIC: 1062.36
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:42 Df Model: 1
Mean Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
mu 6.1500e-04 2.703e-02 2.275e-02 0.982 [-5.237e-02,5.360e-02]
Volatility Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
omega 2.8021e-04 7.842e-04 0.357 0.721 [-1.257e-03,1.817e-03]
alpha[1] 0.0000 3.281e-03 0.000 1.000 [-6.431e-03,6.431e-03]
beta[1] 0.9969 4.602e-03 216.612 0.000 [ 0.988, 1.006]
=============================================================================
Covariance estimator: robust
Fitting EGARCH(1,1) model...
Constant Mean - EGARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: EGARCH Log-Likelihood: -516.859
Distribution: Normal AIC: 1041.72
Method: Maximum Likelihood BIC: 1058.79
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:43 Df Model: 1
Mean Model
==============================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------------
mu -1.7269e-03 8.471e-03 -0.204 0.838 [-1.833e-02,1.488e-02]
Volatility Model
===============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-------------------------------------------------------------------------------
omega -6.3099e-04 6.665e-10 -9.468e+05 0.000 [-6.310e-04,-6.310e-04]
alpha[1] -0.0325 1.378e-05 -2359.576 0.000 [-3.253e-02,-3.248e-02]
beta[1] 1.0000 3.385e-11 2.954e+10 0.000 [ 1.000, 1.000]
===============================================================================
Covariance estimator: robust
Fitting TGARCH(1,1) model...
Constant Mean - GJR-GARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GJR-GARCH Log-Likelihood: -518.640
Distribution: Normal AIC: 1047.28
Method: Maximum Likelihood BIC: 1068.63
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:43 Df Model: 1
Mean Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
mu 6.1404e-04 2.716e-02 2.261e-02 0.982 [-5.262e-02,5.385e-02]
Volatility Model
==============================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------------
omega 2.8007e-04 7.843e-04 0.357 0.721 [-1.257e-03,1.817e-03]
alpha[1] 0.0000 6.746e-03 0.000 1.000 [-1.322e-02,1.322e-02]
gamma[1] -1.4614e-10 1.514e-02 -9.651e-09 1.000 [-2.968e-02,2.968e-02]
beta[1] 0.9969 4.811e-03 207.197 0.000 [ 0.987, 1.006]
==============================================================================
Covariance estimator: robust
Fitting PGARCH(1,1) model...
Constant Mean - TARCH/ZARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: TARCH/ZARCH Log-Likelihood: -522.817
Distribution: Normal AIC: 1055.63
Method: Maximum Likelihood BIC: 1076.98
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:44 Df Model: 1
Mean Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
mu -0.0120 2.747e-02 -0.437 0.662 [-6.584e-02,4.183e-02]
Volatility Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
omega 1.7088e-03 2.605e-03 0.656 0.512 [-3.396e-03,6.814e-03]
alpha[1] 0.0000 2.305e-02 0.000 1.000 [-4.518e-02,4.518e-02]
gamma[1] 0.0295 2.359e-02 1.252 0.211 [-1.671e-02,7.577e-02]
beta[1] 0.9852 1.326e-02 74.291 0.000 [ 0.959, 1.011]
=============================================================================
Covariance estimator: robust
Model Comparisons:
ARCH AIC: 1091.2878891517414, BIC: 1104.0951780028602
GARCH AIC: 1045.2798563631154, BIC: 1062.3562414979406
EGARCH AIC: 1041.7182170275305, BIC: 1058.7946021623557
TGARCH AIC: 1047.2798566243118, BIC: 1068.6253380428432
PGARCH AIC: 1055.6334493925367, BIC: 1076.978930811068
Best model based on BIC: EGARCH
Comparing models based on volatility forecasting...
ARCH - Volatility Forecasting MSE: 0.6419641481631231
GARCH - Volatility Forecasting MSE: 0.6012366262752905
EGARCH - Volatility Forecasting MSE: 0.6013808781523678
TGARCH - Volatility Forecasting MSE: 0.6012364888996955
PGARCH - Volatility Forecasting MSE: 0.6077889091498082
Best model based on volatility forecasting (lowest mse): TGARCH
**********************************************************Processing GBPUSD=X*****************************************************************
Fitting ARCH(1) model...
Constant Mean - ARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: ARCH Log-Likelihood: -461.819
Distribution: Normal AIC: 929.639
Method: Maximum Likelihood BIC: 942.446
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:44 Df Model: 1
Mean Model
==============================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------------
mu -3.4596e-03 2.401e-02 -0.144 0.885 [-5.053e-02,4.361e-02]
Volatility Model
==========================================================================
coef std err t P>|t| 95.0% Conf. Int.
--------------------------------------------------------------------------
omega 0.2693 5.535e-02 4.865 1.147e-06 [ 0.161, 0.378]
alpha[1] 0.2765 0.125 2.217 2.662e-02 [3.207e-02, 0.521]
==========================================================================
Covariance estimator: robust
Fitting GARCH(1,1) model...
Constant Mean - GARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GARCH Log-Likelihood: -396.226
Distribution: Normal AIC: 800.452
Method: Maximum Likelihood BIC: 817.528
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:45 Df Model: 1
Mean Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
mu 0.0213 2.034e-02 1.045 0.296 [-1.860e-02,6.111e-02]
Volatility Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
omega 1.2528e-03 6.924e-04 1.809 7.041e-02 [-1.043e-04,2.610e-03]
alpha[1] 2.8289e-13 1.029e-02 2.750e-11 1.000 [-2.016e-02,2.016e-02]
beta[1] 0.9902 1.317e-02 75.170 0.000 [ 0.964, 1.016]
=============================================================================
Covariance estimator: robust
Fitting EGARCH(1,1) model...
Constant Mean - EGARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: EGARCH Log-Likelihood: -387.853
Distribution: Normal AIC: 783.705
Method: Maximum Likelihood BIC: 800.782
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:46 Df Model: 1
Mean Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
mu 0.0124 9.678e-09 1.281e+06 0.000 [1.239e-02,1.239e-02]
Volatility Model
===============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-------------------------------------------------------------------------------
omega -6.1775e-03 2.369e-12 -2.607e+09 0.000 [-6.177e-03,-6.177e-03]
alpha[1] -0.0427 5.392e-10 -7.921e+07 0.000 [-4.271e-02,-4.271e-02]
beta[1] 0.9983 8.692e-11 1.149e+10 0.000 [ 0.998, 0.998]
===============================================================================
Covariance estimator: robust
Fitting TGARCH(1,1) model...
Constant Mean - GJR-GARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GJR-GARCH Log-Likelihood: -395.415
Distribution: Normal AIC: 800.830
Method: Maximum Likelihood BIC: 822.175
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:46 Df Model: 1
Mean Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
mu 0.0164 2.082e-02 0.790 0.430 [-2.437e-02,5.725e-02]
Volatility Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
omega 1.5388e-03 1.960e-03 0.785 0.432 [-2.303e-03,5.380e-03]
alpha[1] 2.6930e-18 5.383e-02 5.003e-17 1.000 [ -0.105, 0.105]
gamma[1] 0.0178 1.613e-02 1.101 0.271 [-1.386e-02,4.938e-02]
beta[1] 0.9799 6.011e-02 16.303 9.451e-60 [ 0.862, 1.098]
=============================================================================
Covariance estimator: robust
Fitting PGARCH(1,1) model...
Constant Mean - TARCH/ZARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: TARCH/ZARCH Log-Likelihood: -401.060
Distribution: Normal AIC: 812.120
Method: Maximum Likelihood BIC: 833.465
No. Observations: 528
Date: Sat, Sep 07 2024 Df Residuals: 527
Time: 20:05:47 Df Model: 1
Mean Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
mu 9.5323e-03 2.103e-02 0.453 0.650 [-3.168e-02,5.074e-02]
Volatility Model
=============================================================================
coef std err t P>|t| 95.0% Conf. Int.
-----------------------------------------------------------------------------
omega 2.7098e-03 2.587e-03 1.047 0.295 [-2.361e-03,7.780e-03]
alpha[1] 1.2030e-12 3.498e-02 3.439e-11 1.000 [-6.856e-02,6.856e-02]
gamma[1] 0.0545 2.896e-02 1.880 6.005e-02 [-2.304e-03, 0.111]
beta[1] 0.9728 2.386e-02 40.763 0.000 [ 0.926, 1.020]
=============================================================================
Covariance estimator: robust
Model Comparisons: ARCH AIC: 929.63885887227, BIC: 942.4461477233888 GARCH AIC: 800.4518750673487, BIC: 817.5282602021738 EGARCH AIC: 783.7052696374003, BIC: 800.7816547722254 TGARCH AIC: 800.8297879626749, BIC: 822.1752693812062 PGARCH AIC: 812.1196711901466, BIC: 833.465152608678 Best model based on BIC: EGARCH Comparing models based on volatility forecasting... ARCH - Volatility Forecasting MSE: 1.1437886905815706 GARCH - Volatility Forecasting MSE: 1.0010917673953181 EGARCH - Volatility Forecasting MSE: 0.9842808459673964 TGARCH - Volatility Forecasting MSE: 0.9898297338822423 PGARCH - Volatility Forecasting MSE: 0.9995394824228735 Best model based on volatility forecasting (lowest mse): EGARCH
The results show that for USD/ZAR and GBP/USD, EGARCH was the best model based on both the Bayesian Information Criterion (BIC) and volatility forecasting. This suggests that these currency pairs exhibit strong asymmetric volatility, where negative shocks, such as bad news, have a greater impact on volatility than positive shocks, making EGARCH's ability to capture this leverage effect particularly effective for both model selection and volatility prediction. For AUD/USD, while EGARCH was the best model according to BIC, TGARCH performed better in volatility forecasting. This indicates that while EGARCH captured the overall volatility structure well, TGARCH's handling of threshold effects provided a better forecast of future volatility. This may reflect that in the AUD/USD market, negative shocks have a different but more subtle impact on volatility that TGARCH captures more effectively in short-term forecasting.
Question 2¶
In [ ]:
stock_tickers = ['TCS.NS', 'COALINDIA.NS', 'BAJFINANCE.NS']
stock_data = {ticker: yf.download(ticker, start=start_date, end=end_date) for ticker in stock_tickers}
[*********************100%%**********************] 1 of 1 completed [*********************100%%**********************] 1 of 1 completed [*********************100%%**********************] 1 of 1 completed
In [ ]:
for ticker, data in stock_data.items():
print(f"Data for {ticker}")
print(data.head())
Data for TCS.NS
Open High Low Close Adj Close \
Date
2022-08-30 3155.000000 3226.500000 3142.100098 3211.149902 3063.410156
2022-09-01 3190.000000 3190.000000 3121.000000 3131.699951 2987.615967
2022-09-02 3163.000000 3163.000000 3120.300049 3130.399902 2986.375732
2022-09-05 3123.649902 3147.949951 3112.250000 3133.399902 2989.237793
2022-09-06 3135.500000 3140.850098 3106.350098 3127.050049 2983.179932
Volume
Date
2022-08-30 3431525
2022-09-01 3546935
2022-09-02 2052900
2022-09-05 2147912
2022-09-06 1936453
Data for COALINDIA.NS
Open High Low Close Adj Close \
Date
2022-08-30 231.800003 235.500000 230.850006 234.800003 198.011673
2022-09-01 234.350006 236.800003 229.550003 230.300003 194.216736
2022-09-02 231.000000 232.449997 228.199997 229.300003 193.373413
2022-09-05 230.000000 232.350006 228.500000 231.600006 195.313049
2022-09-06 232.699997 234.649994 231.000000 232.699997 196.240707
Volume
Date
2022-08-30 12991363
2022-09-01 9682321
2022-09-02 4599518
2022-09-05 4909470
2022-09-06 5307321
Data for BAJFINANCE.NS
Open High Low Close Adj Close \
Date
2022-08-30 7000.000000 7335.000000 7000.000000 7306.250000 7238.995117
2022-09-01 7228.600098 7333.850098 7142.000000 7181.299805 7115.194824
2022-09-02 7217.100098 7313.000000 7173.450195 7190.350098 7124.162109
2022-09-05 7190.700195 7254.000000 7158.000000 7196.200195 7129.958496
2022-09-06 7228.750000 7255.000000 7105.000000 7119.350098 7053.815430
Volume
Date
2022-08-30 1731381
2022-09-01 1462579
2022-09-02 906370
2022-09-05 707025
2022-09-06 791905
In [ ]:
for stock_name, df in stock_data.items():
returns = calculate_returns(df)
evaluate_and_visualize_models(returns, stock_name)
**********************************************************Processing TCS.NS*****************************************************************
Fitting ARCH(1) model...
Constant Mean - ARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: ARCH Log-Likelihood: -806.307
Distribution: Normal AIC: 1618.61
Method: Maximum Likelihood BIC: 1631.24
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:02 Df Model: 1
Mean Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
mu 0.0795 5.588e-02 1.423 0.155 [-3.000e-02, 0.189]
Volatility Model
========================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------
omega 1.4425 0.250 5.775 7.684e-09 [ 0.953, 1.932]
alpha[1] 0.0429 0.162 0.264 0.792 [ -0.275, 0.361]
========================================================================
Covariance estimator: robust
Fitting GARCH(1,1) model...
Constant Mean - GARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GARCH Log-Likelihood: -806.047
Distribution: Normal AIC: 1620.09
Method: Maximum Likelihood BIC: 1636.93
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:04 Df Model: 1
Mean Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
mu 0.0814 5.421e-02 1.502 0.133 [-2.484e-02, 0.188]
Volatility Model
========================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------
omega 0.9842 0.348 2.831 4.639e-03 [ 0.303, 1.666]
alpha[1] 0.0586 0.149 0.393 0.695 [ -0.234, 0.351]
beta[1] 0.2885 0.259 1.115 0.265 [ -0.219, 0.796]
========================================================================
Covariance estimator: robust
Fitting EGARCH(1,1) model...
Constant Mean - EGARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: EGARCH Log-Likelihood: -805.792
Distribution: Normal AIC: 1619.58
Method: Maximum Likelihood BIC: 1636.42
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:05 Df Model: 1
Mean Model
============================================================================
coef std err t P>|t| 95.0% Conf. Int.
----------------------------------------------------------------------------
mu 0.0756 4.520e-03 16.729 8.061e-63 [6.675e-02,8.447e-02]
Volatility Model
==========================================================================
coef std err t P>|t| 95.0% Conf. Int.
--------------------------------------------------------------------------
omega 0.3340 0.130 2.571 1.013e-02 [7.940e-02, 0.589]
alpha[1] 0.1293 0.211 0.613 0.540 [ -0.284, 0.543]
beta[1] 0.1913 0.318 0.602 0.547 [ -0.432, 0.814]
==========================================================================
Covariance estimator: robust
Fitting TGARCH(1,1) model...
Constant Mean - GJR-GARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GJR-GARCH Log-Likelihood: -805.305
Distribution: Normal AIC: 1620.61
Method: Maximum Likelihood BIC: 1641.65
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:06 Df Model: 1
Mean Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
mu 0.0796 5.711e-02 1.393 0.163 [-3.235e-02, 0.191]
Volatility Model
========================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------
omega 0.7106 2.816 0.252 0.801 [ -4.808, 6.229]
alpha[1] 0.0000 1.202 0.000 1.000 [ -2.357, 2.357]
gamma[1] 0.1027 0.565 0.182 0.856 [ -1.005, 1.211]
beta[1] 0.4819 2.790 0.173 0.863 [ -4.986, 5.950]
========================================================================
Covariance estimator: robust
Fitting PGARCH(1,1) model...
Constant Mean - TARCH/ZARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: TARCH/ZARCH Log-Likelihood: -805.693
Distribution: Normal AIC: 1621.39
Method: Maximum Likelihood BIC: 1642.43
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:06 Df Model: 1
Mean Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
mu 0.0772 5.479e-02 1.408 0.159 [-3.022e-02, 0.185]
Volatility Model
==========================================================================
coef std err t P>|t| 95.0% Conf. Int.
--------------------------------------------------------------------------
omega 1.1743 0.718 1.637 0.102 [ -0.232, 2.581]
alpha[1] 0.0802 0.102 0.784 0.433 [ -0.120, 0.281]
gamma[1] -0.0481 5.002e-02 -0.962 0.336 [ -0.146,4.991e-02]
beta[1] 0.0000 0.569 0.000 1.000 [ -1.115, 1.115]
==========================================================================
Covariance estimator: robust
Model Comparisons:
ARCH AIC: 1618.614274825084, BIC: 1631.240044903374
GARCH AIC: 1620.09410527016, BIC: 1636.9284653745465
EGARCH AIC: 1619.5847850549112, BIC: 1636.4191451592976
TGARCH AIC: 1620.6102132194715, BIC: 1641.6531633499546
PGARCH AIC: 1621.3856440470065, BIC: 1642.4285941774897
Best model based on BIC: ARCH
Comparing models based on volatility forecasting...
ARCH - Volatility Forecasting MSE: 11.579840539839495
GARCH - Volatility Forecasting MSE: 11.594308901283053
EGARCH - Volatility Forecasting MSE: 11.559982619434779
TGARCH - Volatility Forecasting MSE: 11.557878604831322
PGARCH - Volatility Forecasting MSE: 11.559177030413606
Best model based on volatility forecasting (lowest mse): TGARCH
**********************************************************Processing COALINDIA.NS*****************************************************************
Fitting ARCH(1) model...
Constant Mean - ARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: ARCH Log-Likelihood: -980.589
Distribution: Normal AIC: 1967.18
Method: Maximum Likelihood BIC: 1979.80
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:07 Df Model: 1
Mean Model
==========================================================================
coef std err t P>|t| 95.0% Conf. Int.
--------------------------------------------------------------------------
mu 0.1861 7.348e-02 2.533 1.130e-02 [4.212e-02, 0.330]
Volatility Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
omega 2.4183 0.340 7.109 1.173e-12 [ 1.752, 3.085]
alpha[1] 0.2848 0.157 1.816 6.938e-02 [-2.259e-02, 0.592]
===========================================================================
Covariance estimator: robust
Fitting GARCH(1,1) model...
Constant Mean - GARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GARCH Log-Likelihood: -977.805
Distribution: Normal AIC: 1963.61
Method: Maximum Likelihood BIC: 1980.44
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:07 Df Model: 1
Mean Model
==========================================================================
coef std err t P>|t| 95.0% Conf. Int.
--------------------------------------------------------------------------
mu 0.1577 7.219e-02 2.184 2.894e-02 [1.620e-02, 0.299]
Volatility Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
omega 0.8934 0.510 1.751 7.989e-02 [ -0.106, 1.893]
alpha[1] 0.2523 0.140 1.804 7.117e-02 [-2.175e-02, 0.526]
beta[1] 0.5070 0.187 2.707 6.782e-03 [ 0.140, 0.874]
===========================================================================
Covariance estimator: robust
Fitting EGARCH(1,1) model...
Constant Mean - EGARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: EGARCH Log-Likelihood: -975.827
Distribution: Normal AIC: 1959.65
Method: Maximum Likelihood BIC: 1976.49
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:08 Df Model: 1
Mean Model
==========================================================================
coef std err t P>|t| 95.0% Conf. Int.
--------------------------------------------------------------------------
mu 0.1578 6.968e-02 2.265 2.351e-02 [2.126e-02, 0.294]
Volatility Model
==========================================================================
coef std err t P>|t| 95.0% Conf. Int.
--------------------------------------------------------------------------
omega 0.2705 0.135 2.000 4.547e-02 [5.450e-03, 0.536]
alpha[1] 0.3948 0.160 2.469 1.355e-02 [8.138e-02, 0.708]
beta[1] 0.7876 0.101 7.779 7.319e-15 [ 0.589, 0.986]
==========================================================================
Covariance estimator: robust
Fitting TGARCH(1,1) model...
Constant Mean - GJR-GARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GJR-GARCH Log-Likelihood: -973.653
Distribution: Normal AIC: 1957.31
Method: Maximum Likelihood BIC: 1978.35
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:08 Df Model: 1
Mean Model
==========================================================================
coef std err t P>|t| 95.0% Conf. Int.
--------------------------------------------------------------------------
mu 0.1966 7.083e-02 2.776 5.510e-03 [5.777e-02, 0.335]
Volatility Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
omega 0.7662 0.373 2.055 3.989e-02 [3.539e-02, 1.497]
alpha[1] 0.3326 0.187 1.775 7.583e-02 [-3.457e-02, 0.700]
gamma[1] -0.2668 0.175 -1.528 0.126 [ -0.609,7.542e-02]
beta[1] 0.5805 0.155 3.738 1.858e-04 [ 0.276, 0.885]
===========================================================================
Covariance estimator: robust
Fitting PGARCH(1,1) model...
Constant Mean - TARCH/ZARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: TARCH/ZARCH Log-Likelihood: -972.694
Distribution: Normal AIC: 1955.39
Method: Maximum Likelihood BIC: 1976.43
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:09 Df Model: 1
Mean Model
==========================================================================
coef std err t P>|t| 95.0% Conf. Int.
--------------------------------------------------------------------------
mu 0.2099 7.676e-02 2.735 6.246e-03 [5.946e-02, 0.360]
Volatility Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
omega 0.3375 0.190 1.779 7.531e-02 [-3.442e-02, 0.709]
alpha[1] 0.2638 0.128 2.068 3.860e-02 [1.384e-02, 0.514]
gamma[1] -0.1509 0.111 -1.361 0.174 [ -0.368,6.645e-02]
beta[1] 0.6767 0.151 4.485 7.302e-06 [ 0.381, 0.972]
===========================================================================
Covariance estimator: robust
Model Comparisons:
ARCH AIC: 1967.177122648522, BIC: 1979.802892726812
GARCH AIC: 1963.609092941013, BIC: 1980.4434530453993
EGARCH AIC: 1959.6536192708732, BIC: 1976.4879793752596
TGARCH AIC: 1957.3050204839146, BIC: 1978.3479706143978
PGARCH AIC: 1955.3878971832896, BIC: 1976.4308473137728
Best model based on BIC: PGARCH
Comparing models based on volatility forecasting...
ARCH - Volatility Forecasting MSE: 112.95719562081153
GARCH - Volatility Forecasting MSE: 114.66886125352477
EGARCH - Volatility Forecasting MSE: 111.24486967124801
TGARCH - Volatility Forecasting MSE: 110.93134570677911
PGARCH - Volatility Forecasting MSE: 109.8635533889916
Best model based on volatility forecasting (lowest mse): PGARCH
**********************************************************Processing BAJFINANCE.NS*****************************************************************
Fitting ARCH(1) model...
Constant Mean - ARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: ARCH Log-Likelihood: -917.964
Distribution: Normal AIC: 1841.93
Method: Maximum Likelihood BIC: 1854.55
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:09 Df Model: 1
Mean Model
========================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------
mu 6.5384e-03 6.820e-02 9.587e-02 0.924 [ -0.127, 0.140]
Volatility Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
omega 2.1521 0.292 7.360 1.833e-13 [ 1.579, 2.725]
alpha[1] 0.1016 9.375e-02 1.083 0.279 [-8.220e-02, 0.285]
===========================================================================
Covariance estimator: robust
Fitting GARCH(1,1) model...
Constant Mean - GARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GARCH Log-Likelihood: -917.736
Distribution: Normal AIC: 1843.47
Method: Maximum Likelihood BIC: 1860.31
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:10 Df Model: 1
Mean Model
=========================================================================
coef std err t P>|t| 95.0% Conf. Int.
-------------------------------------------------------------------------
mu -3.2361e-03 6.847e-02 -4.726e-02 0.962 [ -0.137, 0.131]
Volatility Model
========================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------
omega 1.4130 1.435 0.984 0.325 [ -1.400, 4.226]
alpha[1] 0.0907 0.102 0.892 0.372 [ -0.108, 0.290]
beta[1] 0.3190 0.649 0.491 0.623 [ -0.954, 1.592]
========================================================================
Covariance estimator: robust
Fitting EGARCH(1,1) model...
Constant Mean - EGARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: EGARCH Log-Likelihood: -916.959
Distribution: Normal AIC: 1841.92
Method: Maximum Likelihood BIC: 1858.75
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:11 Df Model: 1
Mean Model
========================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------
mu 0.0151 7.114e-02 0.213 0.831 [ -0.124, 0.155]
Volatility Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
omega 0.3765 0.242 1.558 0.119 [-9.705e-02, 0.850]
alpha[1] 0.1850 0.113 1.636 0.102 [-3.658e-02, 0.407]
beta[1] 0.5758 0.277 2.075 3.795e-02 [3.201e-02, 1.119]
===========================================================================
Covariance estimator: robust
Fitting TGARCH(1,1) model...
Constant Mean - GJR-GARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: GJR-GARCH Log-Likelihood: -916.589
Distribution: Normal AIC: 1843.18
Method: Maximum Likelihood BIC: 1864.22
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:11 Df Model: 1
Mean Model
========================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------
mu 2.8055e-03 6.775e-02 4.141e-02 0.967 [ -0.130, 0.136]
Volatility Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
omega 1.0436 0.548 1.905 5.683e-02 [-3.031e-02, 2.118]
alpha[1] 0.1625 0.132 1.228 0.220 [-9.696e-02, 0.422]
gamma[1] -0.1385 0.149 -0.933 0.351 [ -0.430, 0.153]
beta[1] 0.4736 0.263 1.803 7.142e-02 [-4.129e-02, 0.989]
===========================================================================
Covariance estimator: robust
Fitting PGARCH(1,1) model...
Constant Mean - TARCH/ZARCH Model Results
==============================================================================
Dep. Variable: Returns R-squared: 0.000
Mean Model: Constant Mean Adj. R-squared: 0.000
Vol Model: TARCH/ZARCH Log-Likelihood: -916.060
Distribution: Normal AIC: 1842.12
Method: Maximum Likelihood BIC: 1863.16
No. Observations: 497
Date: Sat, Sep 07 2024 Df Residuals: 496
Time: 20:10:12 Df Model: 1
Mean Model
========================================================================
coef std err t P>|t| 95.0% Conf. Int.
------------------------------------------------------------------------
mu 0.0248 7.243e-02 0.342 0.732 [ -0.117, 0.167]
Volatility Model
===========================================================================
coef std err t P>|t| 95.0% Conf. Int.
---------------------------------------------------------------------------
omega 0.5995 0.295 2.034 4.195e-02 [2.182e-02, 1.177]
alpha[1] 0.1323 8.629e-02 1.533 0.125 [-3.680e-02, 0.301]
gamma[1] -0.0643 9.608e-02 -0.670 0.503 [ -0.253, 0.124]
beta[1] 0.5379 0.223 2.410 1.596e-02 [ 0.100, 0.975]
===========================================================================
Covariance estimator: robust
Model Comparisons: ARCH AIC: 1841.9278396797808, BIC: 1854.5536097580707 GARCH AIC: 1843.4723447869778, BIC: 1860.3067048913642 EGARCH AIC: 1841.917238364929, BIC: 1858.7515984693155 TGARCH AIC: 1843.1787132736674, BIC: 1864.2216634041506 PGARCH AIC: 1842.120045353808, BIC: 1863.1629954842913 Best model based on BIC: ARCH Comparing models based on volatility forecasting... ARCH - Volatility Forecasting MSE: 32.0599508425907 GARCH - Volatility Forecasting MSE: 32.02554627106747 EGARCH - Volatility Forecasting MSE: 31.92403591329105 TGARCH - Volatility Forecasting MSE: 32.222930254099964 PGARCH - Volatility Forecasting MSE: 31.89102989564405 Best model based on volatility forecasting (lowest mse): PGARCH
The results indicate that different models perform best depending on the evaluation metric (BIC or volatility forecast) for each stock. For Bajaj Finance, while the ARCH model had the lowest BIC, indicating it fits the data well, PGARCH provided the best volatility forecast, suggesting it captures more nuances in volatility dynamics over time. In the case of Coal India, PGARCH was the best model both in terms of BIC and volatility forecasting, meaning it not only fits the data well but also excels at predicting future volatility. Finally, for TCS, the ARCH model had the lowest BIC, suggesting it offers a simpler, better-fitting model, but TGARCH produced better volatility forecasts, likely because TCS exhibits asymmetric volatility patterns that TGARCH is better equipped to capture.